A Highly Predictive Ansatz for Leptonic Mixing and CP Violation

We suggest a simple highly predictive ansatz for charged lepton and light neutrino mass matrices, based on the assumption of universality of Yukawa couplings. Using as input the charged lepton masses and light neutrino masses, the six parameters characterizing the leptonic mixing matrix $V_{PMNS}$, are predicted in terms of a single phase $\phi$, which takes a value around $\phi={\frac{\pi}{2}}$. Correlations among variuos physical quantities are obtained, in particular $V^{PMNS}_{13}$ is predicted as a function of ${\Delta}m^2_{21}$, ${\Delta}m^2_{31}$ and $\sin^2(\theta_{sol})$, and restricted to the range $0.167<|V^{PMNS}_{13}|<0.179$.Comment: 9 pages, 4 figure

Quantum mechanics of a constrained particle and the problem of prescribed geometry-induced potential

The experimental techniques have evolved to a stage where various examples of nanostructures with non-trivial shapes have been synthesized, turning the dynamics of a constrained particle and the link with geometry into a realistic and important topic of research. Some decades ago, a formalism to deduce a meaningful Hamiltonian for the confinement was devised, showing that a geometry-induced potential (GIP) acts upon the dynamics. In this work we study the problem of prescribed GIP for curves and surfaces in Euclidean space $\mathbb{R}^3$, i.e., how to find a curved region with a potential given {\it a priori}. The problem for curves is easily solved by integrating Frenet equations, while the problem for surfaces involves a non-linear 2nd order partial differential equation (PDE). Here, we explore the GIP for surfaces invariant by a 1-parameter group of isometries of $\mathbb{R}^3$, which turns the PDE into an ordinary differential equation (ODE) and leads to cylindrical, revolution, and helicoidal surfaces. Helicoidal surfaces are particularly important, since they are natural candidates to establish a link between chirality and the GIP. Finally, for the family of helicoidal minimal surfaces, we prove the existence of geometry-induced bound and localized states and the possibility of controlling the change in the distribution of the probability density when the surface is subjected to an extra charge.Comment: 21 pages (21 pages also in the published version), 2 figures. This arXiv version is similar to the published one in all its relevant aspect